This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193002 #25 Mar 25 2019 02:32:11
%S A193002 -3,1,0,1,0,3,1,0,2,0,1,0,1,0,-3,1,0,0,0,-5,0,1,0,-1,0,-6,0,3,1,0,-2,
%T A193002 0,-6,0,8,0,1,0,-3,0,-5,0,14,0,-3,1,0,-4,0,-3,0,20,0,-11,0,1,0,-5,0,0,
%U A193002 0,25,0,-25,0,3,1,0,-6
%N A193002 Triangle T(n,k)=0 (k odd), T(0,0)=-3, T(n,0)=1 (n > 0) and T(n,k) = T(n-1,k) - T(n-2,k-2).
%C A193002 Consider an array with recurrence BB(m,n) = BB(m,n-1) + BB(m-1,n), m >= 0:
%C A193002 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,
%C A193002 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7,
%C A193002 3, 5, 6, 6, 5, 3, 0, -4, -9, -15, -22,
%C A193002 3, 8, 14, 20, 25, 28, 28, 24, 15, 0, -22,
%C A193002 3, 11, 25, 45, 70, 98, 126, 150, 165, 165, 143,
%C A193002 3, 14, 39, 84, 154, 252, 378, 528, 693, 858, 1001,
%C A193002 with BB(m,n) = (3m-n)*binomial(n+m-1,n)/m if m > 0. So the BB are polynomials of degree m in n:
%C A193002 BB(1,n) = -(n-3)/1,
%C A193002 BB(2,n) = -(n-6)*(n+1)/2, (see A055999)
%C A193002 BB(3,n) = -(n-9)*(n+1)*(n+2)/6,
%C A193002 BB(4,n) = -(n-12)*(n+1)*(n+2)*(n+3)/24,
%C A193002 BB(5,n) = -(n-15)*(n+1)*(n+2)*(n+3)*(n+4)/120.
%C A193002 Columns in the array are A010701, A016789, A095794, A005564, A059302.
%C A193002 T(n,k) is a zero-padded, column-shifted, sign-modified transpose of this array.
%F A193002 Sum_{k=0..n} T(n,k) = A130806(n+5). (row sums)
%F A193002 Sum_{k=0..n} (-1)^(k/2)*T(n,k) = -A000032(n-2). (alternating row sums)
%F A193002 T(n,k) = (-1)^(1+k/2)*BB(k/2,n-k). - _R. J. Mathar_, Aug 30 2011
%F A193002 T(n,2k) = (-1)^(1+k)*(5-n/k)*binomial(n-k-1,k-1), k > 0. - _R. J. Mathar_, Aug 30 2011
%e A193002 Triangle begins
%e A193002 -3;
%e A193002 1, 0;
%e A193002 1, 0, 3;
%e A193002 1, 0, 2, 0;
%e A193002 1, 0, 1, 0, -3;
%e A193002 1, 0, 0, 0, -5, 0;
%e A193002 1, 0, -1, 0, -6, 0, 3;
%e A193002 1, 0, -2, 0, -6, 0, 8, 0;
%e A193002 1, 0, -3, 0, -5, 0, 14, 0, -3;
%e A193002 1, 0, -4, 0, -3, 0, 20, 0, -11, 0;
%p A193002 BB := proc(m,n) if m=0 then if n= 0 then 3 ; else -1; end if; else (3*m-n)*binomial(n+m-1,n)/m ; end if; end proc:
%p A193002 A193002 := proc(n,k) if type(k,'odd') then 0; else (-1)^(1+k/2)*BB(k/2,n-k) ; end if; end proc:
%p A193002 seq(seq(A193002(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Aug 30 2011
%Y A193002 Cf. A174559.
%K A193002 sign,easy,tabl
%O A193002 0,1
%A A193002 _Paul Curtz_, Jul 14 2011